A249248 → reverse → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-18\,n x^{2} + 9\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-8\,n x^{3} + 4\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-4}'(x) + \left(-8\,n x + 27\,n x^{2} - 4\,n^{2} x + 18\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(8\,n x^{2} + 12\,n x^{3} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-3}'(x) + \left(-2 - 4\,x + 9\,x^{2} - n + 4\,n x + 9\,n x^{2} + 4\,n^{2} x\right)\,P_{n-2}(x) + \left(4\,x^{2} + 4\,x^{3} - 4\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(3 + 4\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 3 + 28\,x + 9\,x^{2}\)

\(P_{3}(x) = 3 + 88\,x + 108\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	3	28	9													40
3	3	88	108	16												215
4	3	160	621	288	25											1097
5	3	244	2169	2544	625	36										5621
6	3	340	5238	14128	7750	1188	49									28696
7	3	448	10395	54288	61625	19548	2058	64								148429
8	3	568	18288	156576	345125	208728	43218	3328	81							775915
9	3	700	29646	373888	1438025	1593108	593390	86656	5103	100						4120619
10	3	844	45279	784224	4713375	9130644	5886909	1482624	161109	7500	121					22212632
11	3	1000	66078	1497168	12949125	40766976	44335494	18472128	3353643	282000	10648	144				121734407
12	3	1168	93015	2661088	31218000	147648096	261385551	176337024	51126228	7008000	469843	14688	169			677962873
13	3	1348	127143	4471056	68126500	453656376	1239188391	1328444736	601605792	128091000	13726603	751248	19773	196		3838210165
14	3	1540	169596	7177488	137481775	1226067336	4871147085	8067999552	5630566851	1817596200	295911913	25474320	1160016	26068	225	22080779968

Technical details

Node id: n-9b509285184957e1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-46a2daf31cb6550f

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 3,   4;
  2: 3,  28,     9;
  3: 3,  88,   108,     16;
  4: 3, 160,   621,    288,      25;
  5: 3, 244,  2169,   2544,     625,      36;
  6: 3, 340,  5238,  14128,    7750,    1188,     49;
  7: 3, 448, 10395,  54288,   61625,   19548,   2058,    64;
  8: 3, 568, 18288, 156576,  345125,  208728,  43218,  3328,   81;
  9: 3, 700, 29646, 373888, 1438025, 1593108, 593390, 86656, 5103, 100;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = (9*n^3+9*n^2-18*n)*x^2*P(n-4,x) + (4*n^3+4*n^2-8*n)*x^3*d/dx P(n-4,x) - ((4*n^2+8*n)*x-(18*n^2+27*n)*x^2)*P(n-3,x) + ((4*n^2+8*n)*x^2+(8*n^2+12*n)*x^3)*d/dx P(n-3,x) - (n+2-(4*n^2+4*n-4)*x-(9*n+9)*x^2)*P(n-2,x) - ((4*n^2+4*n-4)*x^2-(4*n+4)*x^3)*d/dx P(n-2,x) + (2*n+3+(4*n+4)*x)*P(n-1,x) - (4*n+4)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 3 + 28*x + 9*x^2, P(3,x) = 3 + 88*x + 108*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 3 + 28*x + 9*x^2;
P[3, x_] := 3 + 88*x + 108*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 + 9*n^2 - 18*n)*x^2*P[n - 4, x] + (4*n^3 + 4*n^2 - 8*n)*x^3*D[P[n - 4, x], x] - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*P[n - 3, x] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*D[P[n - 3, x], x] - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n + 3 + (4*n + 4)*x)*P[n - 1, x] - (4*n + 4)*x^2*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; v[3] = 3 + 28*x + 9*x^2; v[4] = 3 + 88*x + 108*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 + 9*n^2 - 18*n)*x^2*v[i - 4] + (4*n^3 + 4*n^2 - 8*n)*x^3*deriv(v[i - 4], x) - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*v[i - 3] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*deriv(v[i - 3], x) - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*v[i - 2] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n + 3 + (4*n + 4)*x)*v[i - 1] - (4*n + 4)*x^2*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.